Monoid Category

A monoidal category is a construct in category theory that represents a monoid. It is equipped with an associative binary operation and a neutral element.

In simpler terms, a monoid forms a category with just one object, with each element of the monoid functioning as a morphism.

The key characteristics of a monoid category are:

Objects: This category contains a single unique object, often represented by a generic symbol like $\ast$, which serves as a focal point for defining morphisms.
Note: The "unique" object can be anything—an abstract entity such as a point, an apple, or a pencil. It's typically left undefined to emphasize its abstract nature. For a clearer illustration, one might envision the unique object as a solitary apple.
Morphisms: Each element of the monoid is depicted as a morphism from $\ast$ to $\ast$ within the category. This setup creates a direct correspondence between the monoid's elements and the category’s morphisms.
Morphism Composition: Morphisms combine according to the monoid's binary operation. For example, if $ f $ and $ g $ are morphisms corresponding to the monoid elements $ m $ and $ n $, their composition $ f \circ g $ reflects the element $ m \cdot n $ of the monoid, where "·" denotes the monoid operation.
Note: As a monoid is essentially a semigroup with associative properties, composition inherently satisfies associativity, meaning $(f \circ g) \circ h = f \circ (g \circ h)$ for all morphisms $ f, g, h $.
Identity: The category features an identity morphism, aligned with the monoid’s neutral element. If $ e $ represents this neutral element, then the identity morphism $ \text{id}_{\ast} $ equates to $ e $. Consequently, any morphism $ f $ applied with $ \text{id}_{\ast} $ remains unchanged: $ \text{id}_{\ast} \circ f = f \circ \text{id}_{\ast} = f $.
Note: A monoid integrates an identity element by definition, establishing a baseline or neutral state. In the natural numbers under addition (N,+), this neutral element is zero.

The categorical structure of a monoid category perfectly aligns with that of a monoid as an algebraic structure.

Each monoid property, like associativity and neutral elements, translates directly into equivalent properties within the category, impacting both morphism composition and identity.

Note: Monoid categories are particularly relevant in various fields of mathematics and computer science, including programming language semantics, where they model types with associative and neutral operations.

A Practical Example
Distinguishing Monoidal Categories from Groupoids
Distinguishing Between Monoid Category and Monoidal Category

A Practical Example

Consider the monoid of natural numbers including zero $ N_0 $ with addition as the operation.

$$ (\mathbb{N}_0, + ) $$

This monoid defines a category with a single object, represented by the symbol $\ast$.

The nature of the object is abstract and could be anything, thus it remains undefined.

Note: For an easier understanding, think of the object as an apple. This visual can help clarify the abstract concept.

In the category, morphisms are natural numbers $0, 1, 2, \dots$, each number acting as a morphism for the "adding" operation in an imagined container.

For instance, the number 0 means "add zero objects" (zero apples), 1 signifies "add one object" (one apple), and 2 means "add two objects" (two apples), and so forth.

The composition of two morphisms, $a$ and $b$, is their sum $a + b$, ensuring closure within $ N_0 $ as the result remains within the set of natural numbers.

For example, combining morphisms a=2 (two apples) and b=5 (five apples), their composition a+b=5+2=7 results in another morphism (seven apples) within the category.

The identity is the number zero $0$, which preserves the identity of any morphism it combines with.

For instance, combining the morphism a=2 with the identity morphism e=0 maintains the original morphism: a+e=e+a=a, i.e., 2+0=0+2=2.

This monoid category offers a tangible example of how a monoid's structure is depicted in category theory, clearly illustrating the interplay between algebraic and categorical structures.

Distinguishing Monoidal Categories from Groupoids

Monoidal and groupoid categories cater to different functions and cannot be directly compared. Each category is characterized by unique properties designed to meet distinct needs.

The groupoid category is centered around the principle that every morphism is an isomorphism, thereby making every transformation reversible. This focuses on symmetry and adaptability in the interactions between objects.
The monoidal category introduces a binary operation among objects that is associative and features a unit element. Unlike in a groupoid, isomorphisms are not a given. In a monoidal category, while some morphisms might be isomorphisms, others are not. This category emphasizes the structural aspects brought about by the binary operation and the presence of a unit object.

Therefore, while groupoids emphasize the reversibility of relationships, monoidal categories focus on the additional algebraic structures that enable objects to combine in ways that uphold associativity and identity. These represent two distinct concepts.

Distinguishing Between Monoid Category and Monoidal Category

These terms are frequently confused, yet they represent distinct concepts.

When a monoid forms a category with a single object, we typically refer to it as a monoid category.

This classification stems from the direct correlation between the category's and the monoid's structure:

each monoid element is a morphism
morphism composition mirrors the monoid operation
the neutral element of the monoid acts as the identity morphism for the category's sole object

Note: It's worth noting that not all categories with a single object are monoids, so the distinction is crucial.

Conversely, a monoidal category refers to a more specialized, intricate concept in category theory. It involves a category furnished with a tensor product and a unit object, ensuring properties of associativity and identity are met through natural isomorphisms.

Summarizing:

A monoid category is a category with a single object where morphisms form a monoid under composition. Any monoid can be portrayed as such a category.
A monoidal category includes a tensor product, acting as a binary operation on objects and morphisms, ensuring coherence and associativity up to isomorphism, with an identity object. This broader concept generalizes beyond monoids to encapsulate multiplication and other binary operations in a categorical setting.

While every monoid can indeed form a category known as a "monoid category," not every category derived from a monoid qualifies as a monoidal category unless it also fulfills specific criteria pertaining to monoidal categories.